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On a Generalisation of Finite T-Groups


Let \(\sigma =\{\sigma _i |i\in I\}\) be some partition of all primes \({\mathbb {P}}\) and G a finite group. A subgroup H of G is said to be \(\sigma \)-subnormal in G if there exists a subgroup chain \(H=H_0\le H_1\le \cdots \le H_n=G\) such that either \(H_{i-1}\) is normal in \(H_i\) or \(H_i/(H_{i-1})_{H_i}\) is a finite \(\sigma _j\)-group for some \(j \in I\) for \(i = 1, \ldots , n\). We call a finite group G a \(T_{\sigma }\)-group if every \(\sigma \)-subnormal subgroup is normal in G. In this paper, we analyse the structure of the \(T_{\sigma }\)-groups and give some characterisations of the \(T_{\sigma }\)-groups.

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Correspondence to Chi Zhang.

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Research was supported by the Fundamental Research Funds for the Central Universities (No. 2020QN20) and NSFC of China(No. 12001526)

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Zhang, C., Guo, W. & Liu, AM. On a Generalisation of Finite T-Groups. Commun. Math. Stat. 10, 153–162 (2022).

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  • Finite groups
  • \(\sigma \)-groups
  • Generalised T-groups
  • \(\sigma \)-subnormal
  • The condition \({\mathfrak {R}}_{\sigma _i}\)

Mathematics Subject Classification

  • 20D10
  • 20D15
  • 20D20
  • 20D35